Advanced Probability

Fall 99

Final Examination(Take Home)

Answer all questions Due before Jan 10 (If grades can wait till Jan 25) Due before Dec 20(If grades are needed by Jan 5)

Q1. Using the strong law of large numbers and binary expansion construct a distribution function F (x) such that

(i)F (0) = 1 and F (1) = 1

(ii) F (x) is continuous.

(iii) The measure α on [0, 1] corresponding to F is singular i.e there is set A such that α(A) = 1 while μ(A) = 0 where μ is the Lebesgue measure.

Q2. If X_{n} is the random variable that takes the value ±a_{n} with probability \frac{1}{2} and

σ_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2}

is the variance of S_{n} = X_{1} + ⋯ + X_{n} we write

\frac{S_{n}}{σ_{n}} = X_{n,1} + ⋯ + X_{n,n}

When is X_{n,j} uniformly infinitesimal? When is the Lindeberg condition satisfied? Express your answers as conditions on the sequence a_{n} ? Does a_{n} = 2^{n} or a_{n} = 2^{−n} or a_{n} = n satisfy them?

Q3. The law of the iterated logarithm states that if X_{1}, X_{2}, ⋯ are independent identically distributed random variables mean 0 and variance 1

lim sup_{n→∞} \frac{S_{n}}{\sqrt{2n log log n}} = +1

lim inf_{n→∞} \frac{S_{n}}{\sqrt{2n log log n}} = −1

with probability 1. Does this mean that \frac{1}{2} is necessairly a limit point? Why?

Q4. {π_{i,j}} are the transition probabilities of an irreducible Markov chain on the positive integers i ≥ 1. If there are numbers pi > 0 such that

∑_{i} p_{i}π_{i,j} = p_{j}

for every j, show that the chain is positive recurrent if and only if

∑_{j} p_{j} < ∞.

Q5. Use the criterion of Q4. to determine the values of the parameters 0 < α < 1 and β for which the following chain is positive recurrent.

π_{i,j} = \{ \begin{matrix}α + \frac{β}{i+4} & if j = 1 + 1 and i is large \\ 1 − π_{i,i+1} & if j = i − 1 and i is large \\ 0 & if j ≠ i ± 1 with i large \end{matrix}

with π_{i,j} defined arbitrarily for small i so that the chain is irreducible and π_{i,j} = 0 unless |j − i| ≤ 1